"In the Great Pyramid the Egyptians produced a system of map
projection even more sophisticated than the one incorporated in the
ziggurats. The apex of the Pyramid corresponds to the pole, the
perimeter to the equator, both in proper scale. Each flat face of the
Pyramid was designed to represent one curved quarter of the northern
hemisphere, or sherical quadrant of 90 degrees.
To project a spherical quadrant onto a flat triangle correctly,
the arc, or base, of the quadrant must be the same length as the base
of the triangle, and both must have the same height. This happens to be
the case only with a cross section or meridian bisection of the Great
Pyramid, whose slope anle gives the pi relation between height and
base.
The subtlety of the Pyramid's projection lies in the fact that
when viewed from the side, the laws of perspective reduce the actual
area of a face (mathematically oversized) to the correct size for the
projection, which is the Pyramid's cross section. What the viewer saw,
and sees, with the aid of perspective is the correct triangle. The key
to the geometrical and mathematical secret of the Pyramid, so long a
puzzle to mankind, was actually handed to Herodotus by the temple
priests (Heliopolis) when they informed him that the Pyramid was
designed in such a way that the area of each of its faces was equal to
the square of its height.
This interesting observation reveals that the Pyramid was
designed to incorporate not only the pi proportion but another and even
more useful constant proportion, known in the Renaissance as the Golden
Section, designated in modern times by the Greek letter phi, or 1.618.
(If the 356 cubits of the Pyramid's apothem are divided by half the
base, or 220 cubits, the result is 89/55, or 1.618.) Phi, like pi,
cannot be worked out arithmetically; but it can easily be obtained with
nothing more than a compass and a straightedge. With the incorporation
of the Golden Section, the Great Pyramid provides an effective system
for translating sperical areas into flat ones.
John Taylor set about drawing and redrawing every feature of
the Pyramid on the basis of the measurements reported by Howard-Vyse,
so as to see what geometrical or mathematical formulas might be derived
from the structure. Taylor was puzzled as to why the builders of the
Pyramid should have chosen the particular anle of 51 degrees 51' for
the Pyramid's faces instead of the regular equilateral triangle of 60
degrees. Analyzing Herodotus' report of what the Egyptian priest had
told him about the surface of each face of the Pyramid, Taylor
concluded they had been designed to be equal in area to the square of
the Pyramid's height. If so, this meant the building was of a
particular if not unique geometric construction; no other pyramid has
these proportions.
Taylor then discovered that if he divided the perimeter of the
Pyramid by twice its height, it gave hem a quotient of 3.144,
remarkable close to the value of pi, which is computed as 3.14159+. In
other words, the height of the Pyramid appeared to be in relation to
the perimeter of its base as the radius of a circle is to its
circumference. This seemed to Taylor far too extraordinary to attribute
to chance, and he deduced that the Pyramid might have been specifically
intended by its builders to incorporate the incommensurable value of
pi. If so, this was a demonstration of the advanced knowledge of the
builders. Still today the oldest known document which indicates that
the Egyptians had a knowledge of the value of pi is the Rhind Papyrus,
dated about 1700 B.C., and therefore much later than the Pyramid. Found
in the wrappings of a mummy in 1855 by a young Scottish archeologist,
Henry Alexander Rhind, the rare papyrus is now in the British Museum.
It gives a very rough value for pi of 3.16.
Searching for a reason for such a pi proportion in the Pyramid,
Taylor concluded that the perimeter might have been intended to
represent the circumference of the earth at the equator while the
height represented the distance from the earth's center to the pole."
(Which, with today's technology, the Pyramid has recently been
remeasured and computed in 1994 proving this to be true.)